Transcript Wednesday, Feb. 11, 2015

PHYS 3313 – Section 001
Lecture #7
Wednesday, Feb. 11, 2015
Dr. Jaehoon Yu
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Wednesday, Feb. 11,
2015
Invariant Quantities
The Doppler Effect
Relativistic Momentum and Energy
Relationship Between Relativistic
Quantities
Quantization
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
1
Announcements
• Reading assignments: CH 3.3 (special topic –
the discovery of Helium) and CH3.7
• Colloquium today Dr. William Newton of TAMU
Commerce
– Into the Mantle of the Neutron Star: Probing Condensed
Nuclear Matter under Extreme Condition
– SH101 at 4pm
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
2
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4.
Reminder: Special Project #2
Derive the three Lorentz velocity transformation equations, starting from
Lorentz coordinate transformation equations. (10 points)
Derive the three reverse Lorentz velocity transformation equations,
starting from Lorentz coordinate transformation equations. (10 points)
Prove that the space-time invariant quantity s2=x2-(ct)2 is indeed
invariant, i.e. s2=s’2, in Lorentz Transformation. (5 points)
You must derive each one separately starting from the Lorentz spatial
coordinate transformation equations to obtain any credit.
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Just simply switching the signs and primes will NOT be sufficient!
Must take the simplest form of the equations, using and .
5.
You MUST have your own, independent answers to the above three
questions even if you worked together with others. All those who share
the answers will get 0 credit if copied.
• Due for the submission is this Monday, Feb. 16!
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
3
Space-time Invariants
There are three possibilities for the invariant quantity Δs2:
Ds = Dx - D ( ct )
2
2
2
1
3
1) Δs2 = 0: Δx2 = c2 Δt2: light-like separation
– Two events can be connected only by a light signal.
2
2) Δs2 > 0: Δx2 > c2 Δt2: space-like separation
– No signal can travel fast enough to connect the two events.
The events are not causally connected!!
3) Δs2 < 0: Δx2 < c2 Δt2: time-like separation
– Two events can be causally connected.
– These two events cannot occur simultaneously!
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
4
The Doppler Effect
• The Doppler effect of sound
– Increased frequency of sound as a source approaches a receiver
– Decreased frequency as the source recedes.
• The same change in sound frequency occurs when the
source is fixed and the receiver is moving.
– The change in frequency of the sound wave depends on whether
the source or the receiver is moving.
• Does this violate the principle of relativity?
– No
– Why not?
– Sound wave is in a special frame of media such as air, water, or
a steel plate in order to propagate;
• Light does not need a medium to propagate!
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
5
Recall the Doppler Effect
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
6
The Relativistic Doppler Effect
Consider a source of light (a star) and a receiver (an astronomer)
approaching one another with a relative velocity v.
1) Consider the receiver in system K and the light source in
system K’ moving toward the receiver with velocity v.
2) The source emits n waves during the time interval T.
3) Because the speed of light is always c and the source is
moving with velocity v, the total distance between the front
and rear of the wave emitted during the time interval T is:
Length of wave train = cT − vT
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
7
The Relativistic Doppler Effect (con’t)
Because there are n waves emitted by the source in
time period T, the wavelength measured by the
stationary receiver is
cT - vT
l=
n
cn
c
f= =
l cT - vT
And the resulting frequency measured by the
receiver is
The number of waves emitted in the moving frame
of the source is n=f0T ‘0 with the proper time T’0=T/
we obtain the measured frequency by the receiver
as
1- b 2
1 f0
cf0T g
f0 =
=
f=
=
1- b
cT - vT 1- v c g
Wednesday, Feb. 11,
2015
(1- b ) (1+ b ) f =
0
2
(1- b )
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
1+ b
f0
1- b
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Results of Relativistic Doppler Effect
When source/receiver is approaching with
1+ b
f=
f0
β = v/c the resulting frequency is
1- b
Higher than the actual source’s frequency, blue shift!!
1- b
f=
f0
1+ b
When source/receiver is receding with
β = v/c the resulting frequency is
Lower than the actual source’s frequency, red shift!!
If we use +β for approaching
source/receiver and -β for receding
source/receiver, relativistic Doppler
Effect can be expressed
For more generalized case
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
1+ b
f=
f0
1- b
f cosq =
f0 ( cosq ¢ + b )
1- b 2
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Relativistic Momentum
The most fundamental principle used here is the momentum conservation!
Frank is at rest in system K holding a ball of mass m.
Mary holds a similar ball in system K’ that is moving in the x direction with
velocity v with respect to system K.
At one point they threw the ball at each other with exactly the same speed
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
10
Relativistic Momentum
• If we use the definition of momentum, the momentum of the
ball thrown by Frank is entirely in the y direction
pFy = mu0
• The change of momentum as observed by Frank is
ΔpF = ΔpFy = −2mu0
• Mary measures the initial velocity of her own ball to be
u’Mx = 0 and u’My = −u0.
• In order to determine the velocity of Mary’s ball as
measured by Frank we use the velocity transformation
equations:
2
2
uMx = v
Wednesday, Feb. 11,
2015
uMy = -u0 1- v c
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
11
Relativistic Momentum
Before the collision, the momentum of Mary’s ball as measured by Frank (in
the Fixed frame) with the Lorentz velocity transformation becomes
pMx = mv
pMy = -mu0 1- v 2 c 2
For a perfectly elastic collision, the momentum after the collision is
pMx = mv
pMy = +mu0 1- v 2 c 2
Thus the change in momentum of Mary’s ball according to Frank is
DpM = DpMy = 2mu0 1- b ¹ -DpFy
2
OMG! The linear momentum is not conserved even w/o an external force!!
d (g u r )
What do we do?
p= m
= mg u u
Redefine the momentum in a fashion
dt
Something has changed. Mass is now, m!! The relativistic mass!!
Mass as the fundamental property of matter is called the “rest mass”, m0!
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
12
Relativistic and Classical Linear Momentum
Wednesday, Feb. 11,
2015
PHYS 3313-001, Spring 2014
Dr. Jaehoon Yu
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